Friedrich Schmid

University of Cologne | UOC · Seminar für Wirtschafts- und Sozialstatistik

We propose an extension of the univariate Lorenz curve and of the Gini coefficient to the multivariate case, i.e., to simultaneously measure inequality in more than one variable. Our extensions are based on copulas and measure inequality stemming from inequality in each single variable as well as inequality stemming from the dependence structure of...

We propose an extension of the univariate Lorenz curve and of the Gini coefficient to the multivariate case, i.e., to simultaneously measure inequality in more than one variable. Our extensions are based on copulas and measure inequality stemming from inequality in each single variable as well as inequality stemming from the dependence structure of...

We propose an extension of the univariate Lorenz curve and of the Gini coefficient to the multivariate case, i.e., to simultaneously measure inequality in more than one variable. Our extensions are based on copulas and measure inequality stemming from inequality in every single variable as well as inequality stemming from the dependence structure o...

Nonparametric estimation of copula-based measures of multivariate association in a continuous random vector X=(X 1, …, X d ) is usually based on complete continuous data. In many practical applications, however, these types of data are not readily available; instead aggregated ordinal observations are given, for example, ordinal ratings based on a...

Pearson's correlation coefficient is typically used for measuring the dependence structure of stock returns. Nevertheless, it has many shortcomings often documented in the literature. We suggest to use a conditional version of Spearman's rho as an alternative dependence measure. Our approach is purely nonparametric and we avoid any kind of model mi...

Mutual information (also known as Kullback–Leibler divergence) can be viewed as a measure of multivariate association in a random vector. The definition incorporates the joint density as well as the marginal densities. We will focus on a representation of mutual information in terms of copula densities that is thus independent of the marginal distr...

A general, copula-based framework for measuring the dependence among financial time series is presented. Particular emphasis is placed on multivariate conditional Spearman's rho (MCS), a new measure of multivariate conditional dependence that describes the association between large or extreme negative returns—so-called tail dependence. We demonstra...

This paper suggests five measures of association between two random vectors X = (X_1, ..., X_p) and Y = (Y_1, ..., Y_q). They are copula based and therefore invariant with respect to the marginal distributions of the components X_i and Y_j. The measures capture positive as well as negative association of X and Y. In case p = q = 1 they reduce to Sp...

In their seminal article, Kendall and Babington Smith (19399. Kendall , M. G. , Babington Smith , S. B. ( 1939 ). The problem of m rankings . The Ann. Mathemat. Statist. 10 : 275 – 287 . [CrossRef]View all references) suggested a measure 𝒲 to quantify the agreement between d rankings of n objects. Its distribution was essentially investigated und...

Das Buch bietet eine Einführung in die wichtigsten Methoden der Wahrscheinlichkeitsrechnung und des statistischen Schließens, das heißt, der Schätzung von Parametern und des Testens von Hypothesen. Die Darstellung zielt auf klare Begriffe, nachvollziehbare Verfahren und Motivation aus den Wirtschaftswissenschaften. Sie enthält zahlreiche durchgerec...

A multivariate measure of association is proposed, which extends the bivariate copula-based measure Phi-Square introduced by Hoeffding [22]. We discuss its analytical properties and calculate its explicit value for some copulas of simple form; a simulation procedure to approximate its value is provided otherwise. A nonparametric estimator for multi...

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical...

This chapter constitutes a survey on copula-based measures of multivariate association - i.e. association in a d-dimensional random vector X = (X1 , ¼,Xd )X = (X_1 , \ldots ,X_d ) where d ³ 2d \ge 2 . Some of the measures discussed are multivariate extensions of wellknown bivariate measures such as Spearman’s rho, Kendall’s tau, Blomqvist’s beta...

Sharpe ratios (Sharpe 1966) are the most popular risk-adjusted performance measure for investment portfolios and investment funds. Given a riskless security as a benchmark, its Sharpe ratio is defined by $$SR = \frac{{\mu - z}}{{\sqrt {{\sigma ^2}} }}$$ where μ and σ2 denote the portfolio’s mean return and return volatility, respectively, and z rep...

A goodness of fit test for copulas based on Rosenblatt's transformation is investigated. This test performs well if the marginal distribution functions are known and are used in the test statistic. If the marginal distribution functions are unknown and are replaced by their empirical estimates, then the test's properties change significantly. This...

This paper provides a survey on studies that analyze the macroeconomic effects of intellectual property rights (IPR). The first part of this paper introduces different patent-policy instruments and reviews their effects on R&D and economic growth. This part also discusses the distortionary effects and distributional consequences of IPR protection a...

A new family of conditional-dependence measures based on Spearman's rho is introduced. The corresponding multidimensional versions are established. Asymptotic distributional results are derived for related estimators which are based on the empirical copula. Particular emphasis is placed on a new type of multidimensional tail-dependence measure and...

Multivariate measures of association are considered which, in the bivariate case, coincide with the population version of Spearman's rho. For these measures, nonparametric estimators are introduced via the empirical copula. Their asymptotic normality is established under rather weak assumptions concerning the copula. The asymptotic variances are ex...

We examine the effect of political [`]institutions' on economic growth volatility, using data from more than 100 countries over the period 1960 to 2005, taking into account various control variables as suggested in previous studies. Our indicator of volatility is the relative standard deviation of the growth rate of GDP per capita. The results of a...

Spearman's rho can be generalized to the multivariate, i.e. d-dimensio-nal case. Nonparametric estimation of Spearman's multivariate rho has recently been considered and the asymptotic normality for the estimator has been established. Though a closed and compact formula for the asymptotic variance exists, it is not suitable for practical applicatio...

Bei Finanzzeitreihen beobachtet man im Zeitablauf wechselnde Volatilitäten: Phasen hoher Volatilität wechseln sich ab mit Phasen geringer Volatilität. Man bezeichnet diese Eigenschaft auch als bedingte Heteroskedastizität. Mit den in Kapitel 4 eingeführten linearen ARMA-Prozessen kann man sie nicht modellieren, vielmehr sind nichtlineare Prozesse w...

Das Capital Asset Pricing Modell (CAPM) wurde von Sharpe, Lintner und Mossin Mitte der 60er Jahre entwickelt. Es ist ein Gleichgewichtsmodell, das unter (sehr restriktiven) Annahmen die Preisbildung risikobehafteter Finanztitel erklärt und wichtige Folgerungen über die Beziehung von erwarteter Rendite und Risiko von Wertpapieren zulässt. Obwohl das...

Im Mittelpunkt dieses Kapitels steht die univariate Verteilung der Renditen eines Wertpapiers. Wir erinnern zunächst an einige Grundbegriffe aus der Wahrscheinlichkeitsrechnung wie z.B. Zufallsvariable, Verteilungsfunktion und Dichtefunktion. Wir beschreiben Verfahren, mit denen die Renditeverteilung aus beobachteten Renditen geschätzt und graphisc...

Im vorangegangenen Kapitel wurden univariate Renditeverteilungen betrachtet; es wurde jeweils nur die Rendite eines einzelnen Finanzmarkttitels untersucht. Aus verschiedenen Gründen ist es jedoch oft erforderlich, die Renditen mehrerer Finanztitel simultan zu betrachten. Hat man ein Portfolio aus mehreren Wertpapieren, so spielt die Abhängigkeit de...

Der Begriff der Informationseffizienz und die damit verbundene Random-Walk-Hypothese ist grundlegend für die Kapitalmarkttheorie. Wir behandeln sie in Abschnitt 5.1. In den nachfolgenden Abschnitten werden dann verschiedene Tests vorgestellt, mit denen die Random-Walk-Hypothese empirisch überprüft werden kann.

Dieses Buch hat die Beschreibung und Analyse von Finanzmarktdaten, also von Kursen und vor allem Renditen von Wertpapieren, zum Inhalt. Dieses Kapitel behandelt die wichtigsten Grundbegriffe. Abschnitt 1.1 beschäftigt sich mit Kursen und deren Bestimmung. Abschnitt 1.2 behandelt Renditedefinitionen und diskutiert deren Eigenschaften. Abschnitt 1.3...

This article suggests a chi-square test of fit for parametric families of bivariate copulas. The marginal distribution functions are assumed to be unknown and are estimated by their empirical counterparts. Therefore, the standard asymptotic theory of the test is not applicable, but we derive a rule for the determination of the appropriate degrees o...

Third degree stochastic dominance is usually defined by stating two separate conditions. It is shown in this note that the second condition is superfluous, because it is already implied by the first one. http://link.springer.com/article/10.1007%2Fs00291-005-0208-8

The lower tail dependence λL is a measure that characterizes the tendency of extreme co-movements in the lower tails of a bivariate distribution. It is invariant with respect to strictly increasing transformations of the marginal distribution and is therefore a function of the copula of the bivariate distribution. λL plays an important role in mode...

Commonly used standard statistical procedures for means and variances (such as the t–test for means or the F–test for variances and related confidence procedures) require observations from independent and identically normally distributed variables. These procedures are often routinely applied to financial data, such as asset or currency returns, wh...

The pollution-convergence hypothesis is formalized in a neoclassical growth model with optimal emissions reduction: pollution growth rates are positively correlated with output growth (scale effect) but negatively correlated with emission levels (defensive effect). This dynamic law is empirically tested for two major and regulated air pollutants -...

We consider the classical testing problem H 0 :F=G against H 1 :F≤G, F≠G, for continuous distribution functions F and G, when data is available as matched pairs (X i ,Y i ), i=1,⋯,n. The power of tests suggested by Z. Govindarajulu [Studia Sci. Math. Hungar. 10(1975), 39–45 (1977; Zbl 0362.62054)] and by U. Munzel [Stat. Neerl. 53, 277–286 (1999; Z...

Area statistics are sample versions of areas occurring in a probability plot of two distribution functions F and G. This paper presents a unified basis for five statistics of this type. They can be used for various testing problems in the framework of the two sample problem for independent observations, such as testing equality of distributions aga...

This note investigates the structure of dominance relations in daily returns of 32 German assets during the 1990s. We focus on stochastic dominance of first, second and third order but mean-variance and mean-Gini dominance is also considered. Efficient (i. e., nondominated) sets of assets are determined with respect to the various criteria. The inv...

This note investigates the structure of dominance relations in daily returns of 32 German assets during the 1990s. We focus on stochastic dominance of first, second and third order but mean-variance and mean-Gini dominance is also considered. Efficient (i. e., nondominated) sets of assets are determined with respect to the various criteria. The inv...

Area statistics are sample versions of areas occuring in a probability plot of two distribution functions F and G. This paper gives a unified basis for five statistics of this type. They can be used for various testing problems in the framework of the two sample problem for independent observations such as testing equality of distributions against...

This paper revisits finite normal mixtures as models for the distribution of daily stock returns. Estimates for parameters of finite mixtures can easily be derived from the well known EM-algorithm which turns out to be numerically stable for the type of data under study. For 2-component mixtures the estimates are presented in detail for 30 german s...

A nonparametric test for second-order stochastic dominance is introduced in the framework of the one sample problem. It is based on a supremum statistic which is suitable for second-order problems. Its asymptotic distribution is identified and quantiles of the finite sample and asymptotic distributions are derived by simulations. Its power is compa...

Power of modifications of the Kolmogorov, Cramer-von Mises, Watson and Anderson-Darling tests for testing uniformity when limits are unknown is compared. Power is computed by Monte Carlo simulation within one-parameter families of alternative distributions containing the uniform distribution as a special case. A table of mostly unpublished quantile...

A χ2-test of fit for testingH 0 “X∼U(a,b), a,b unknown” is suggested. It is nonstandard because the usual regularity assumptions are not satisfied. The asymptotic distribution of the test statistic underH 0 is derived. The error probabilities of the first kind are investigated by Monte Carlo simulation for samples of small and medium size. http:/...

Modifications of the Kolmogorov, Cramér-von Mises, and Watson tests for testing uniformity with unknown limits are suggested. It is shown that the asymptotic distributions of the modified test statistics coincide with those of the respective standard versions. Precise tables of the quantiles of the modified statistics are obtained by Monte Carlo si...

This paper deals with nonparametric inference for second order stochastic dominance of two random variables. If their distribution functions are unknown they have to be inferred from observed realizations. We establish two methods to take the sampling error into account. The first one is based on the asymptotic normality of point estimators, while...

An L1-variant of the Cramer-von Mises test statistic for the one sample test of fit problem is presented. Quantiles of the sampling distribution under the null hypothesis are derived by Monte-Carlo Simulation. The power of the new test is compared to those of other, conventional one sample tests, such as the Cramer-von Mises test, the Anderson-Darl...

This paper introduces a new distribution-free test for first-order stochastic dominance. It can be viewed as a modification of the well-known Wilcoxon-Mann-Whitney test. As it is based on the pp-plot of two distribution functions the new test statistic has a nice pictorial interpretation. Its finite sample distribution as well as its asymptotic dis...

The finite sample distribution and the asymptotic distribution of a test statistic originally suggested by Mosler (1995) for testing H0 : " F(x) ≤ G(x) for x ∈ ℝ or G(x) ≤ F(x) for x 6 ∈ ℝ" is derived and some relevant quantiles are tabulated. The power of the corresponding test is investigated for selected alternatives.

A new distribution free test for the two sample problem is presented. The test statistic is derived from a descriptive measure of interdistributional inequality, thus having an intuitive pictorial interpretation. Some quantiles of the exact distribution of the new test statistic under H0 are computed for balanced samples sized up to fourteen observ...

The paper surveys various parametric Lorenz curves to be fitted to ground income data in order to obtain an estimate for the Gini measure of inequality. The curves are fitted to 16 sets of empirical income data. The results are compared to the results of the purely nonparametric method (due to Gastwirth) of computing lower and upper bounds for the...

Economic data are often given in grouped form, i.e., only group limits, group frequencies and group means are reported. We derive distribution free lower and upper bounds for shares and quantiles from this type of data.

It is common practice to summarize income or wealth inequality in a population by computing statistical inequality measures like the Gini Coefficient or the Coefficient of Variation. This should be complemented, however, by an analysis of the structure of inequality and an attempt to identify its sources. For this purpose it may be useful to partit...

Measurement of poverty in the spirit of Sen essentially consists of two steps, namely (1) the identification of the poor and (2) the aggregation of the income distribution of the poor into an indicator called poverty measure. We contribute to the second step from a mathematical and statistical point of view by defining a class of poverty measures a...

Small sample properties of seven confidence intervals for the binomial parameterp (based on various normal approximations) and of the Clopper-Pearson interval are compared. Coverage probabilities and expected lower and upper limits of the intervals are graphically displayed as functions of the binomial parameterp for various sample sizes. http://d...

Statistische Daten sind oft nur in gruppierter Form verfgbar. In dieser Arbeit wird ein berblick ber die Maximum-Likelihood-Methode zur Parameterschtzung aus gruppierten Daten gegeben. Nach Einfhrung der Notation wird gezeigt, wie der Informationsverlust, der durch die Gruppierung entsteht, quantifiziert werden kann. Die stochastischen Eigenschafte...

Small sample bias and variance of the ML estimators of the parameters μ, and σ of grouped observations from a normal distribution are investigated using Monte Carlo simulation. The usefulness of the estimates of the variance of ML estimators which can be taken from the last iteration of the Scoring and Newton-Raphson algorithm is shown. In addition...

The effect of various estimation methods on the level of the Chi-square goodness-of-fit test for a negative binomial distribution is investigated. It is shown by Monte-Carlo-Simulation that Method of Moment estimates strongly increase the level of the test whereas Maximum Likelihood estimates from grouped data (i. e. a correct use of the test) resu...

Programs in Standard-Fortran for the computation of the distribution function and percentage points of the central and noncentral F-distribution are given. http://download.springer.com/static/pdf/886/art%253A10.1007%252FBF02932556.pdf?auth66=1413379781_0b51158c72a8832f0474a712ea6326a9&ext=.pdf

The paper shows that One Way ANOVA is highly nonrobust with respect to positive within group correlation because the true level of significance α* of the procedure is much higher than the prescribed level α. Non robustness is also stressed by power considerations.

Computation of M. L. estimates for the parameters of a negative binomial distribution from grouped data is considered. For this problem the Scoring, Newton—Raphson and E-M algorithm is derived. Using simulated data the performance of the algorithms is compared with respect to convergence, number of iterations and computing time. Finally an empirica...

Computation of Maximum Likelihood Estimates for μ and β from a grouped sample of a normal population is a numerical problem which occurs frequently in applied statistics. We investigate the performance of several algorithms for this problem (including Newton-Raphson, Method of Scoring, Expectation-Maximization and others) using simulated data. One...

http://link.springer.com/article/10.1007%2FBF02932735

Zusammenfassung Es wird das allgemeine nichtlineare Regressionsmodell mit stochastischen Regressoren und additivem Störterm betrachtet. Es wird gezeigt, daß unter schwachen Voraussetzungen die Folge der Kleinste-Quadrate-Schätzer stark konsistent ist. Die Arbeit verallgemeinert Ergebnisse von Jennrich und Malinvaud.